Affine Hypersurfaces with Parallel Shape Operator

Affine Hypersurfaces with Parallel Shape Operator We prove the following: a relative hypersurface with parallel shape operator is either a relative hypersphere, or it is affinely equivalent to an example constructed by Th. Binder. Furthermore, based on Binder’s example, we give another simple and more explicit example; this way we improve the classification and show that it is completely determined by functions $$\kappa (t)$$ κ ( t ) and $$C(v_i;t)$$ C ( v i ; t ) , the latter being solutions of certain Monge–Ampère equations. Our example geometrically is constructed from a plane curve and a family of relative hyperspheres. In case of an affine sphere with Blaschke geometry we show that our classification can be considered as a construction coming from a plane curve together with a family of improper affine hyperspheres. Especially in $$\mathbb R^4$$ R 4 , this construction is determined by only three functions of a single variable. Results in Mathematics Springer Journals

Affine Hypersurfaces with Parallel Shape Operator

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Springer International Publishing
Copyright © 2016 by Springer International Publishing
Mathematics; Mathematics, general
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